Gradient Structure of Dynamics for Nonautonomous Cascade Systems

In this paper we prove the gradient structure of solutions for a nonautonomous cascade system defined on Banach spaces, where the x –variable evolves independently via ẋ = Ax+f(t,x) x ˙ = A x + f ( t , x ) and influences the y –variable through ẏ = By+g(x,y) y ˙ = B y + g ( x , y ) . By first analyzing the long–time dynamics of the nonautonomous x –equation and then examining the resulting y –dynamics for each asymptotic state of x , we provide a complete description of the system’s gradient structure in two levels: a more abstract and general, with less hypotheses on f , and a deeper level of description, when the term f ( t , x ) is asymptotically autonomous. Finally, we present a description when the term f ( t , x ) is a small nonautonomous perturbation of an autonomous term.